On small sets of integers

An upper quasi-density on H (the integers or the non-negative integers) is a real-valued subadditive function μ⋆ defined on the whole power set of H such that μ⋆(X) ≤ μ⋆(H) = 1 and μ⋆(k·X+h)=1kμ⋆(X) for all X⊆ H, k∈ N+, and h∈ N, where k· X: = { kx: x∈ X} ; and an upper density on H is an upper quasi-density on H that is non-decreasing with respect to inclusion. We say that a set X⊆ H is small if μ⋆(X) = 0 for every upper quasi-density μ⋆ on H. Main examples of upper densities are given by the upper analytic, upper Banach, upper Buck, and upper Pólya densities, along with the uncountable family of upper α-densities, where α is a real parameter ≥ - 1 (most notably, α= - 1 corresponds to the upper logarithmic density, and α= 0 to the upper asymptotic density). It turns out that a subset of H is small if and only if it belongs to the zero set of the upper Buck density on Z. This allows us to show that many interesting sets are small, including the integers with less than a fixed number of prime factors, counted with multiplicity; the numbers represented by a binary quadratic form with integer coefficients whose discriminant is not a perfect square; and the image of Z through a non-linear integral polynomial in one variable.